From Fenway Park to Logan Airport, the best way is to take the Green Line MBTA to South Station and ride the Blue Line MBTA to Logan Airport.

South Station's on the Red Line. You want to take the Green Line to Government Center and then the Blue Line to the airport. (Government Center stop noted by experience on the MBTA Green Line.)

Edited to add that I've never actually seen anywhere that you can connect to the Blue Line. Although the MBTA site says it's open, I don't think I've seen a Blue Line train at the Government Center station.